Over the past half-century, a lineage of increasingly continuous generalizations of Conway’s Game of Life has beendeveloped—from Larger than Life (Griffeath 1994; Evans 1996) through SmoothLife (Rafler 2011) to Lenia (Chan2019, 2020)—culminating in systems that produce moving, self-repairing, lifelike creatures using unimodal growthfunctions and distance-dependent kernels. Yet the theoretical question of why unimodal growth functions arenecessary has not been addressed. We provide the answer in two theorems. Theorem 1 (Impossibility): for anyStuart-Landau oscillator with linear coupling on a lattice, the steady-state amplitude R*(n) is monotonic in neighborcount n. Theorem 2 (Unimodal Condition): the minimal coupling structure for Life-like rules is a peaked growthfunction of local density—precisely the structure Lenia employs. We frame these results within a four-levelhierarchical decomposition of emergent life: (1) synchronization, (2) cluster formation, (3) density-dependentsurvival, (4) pattern dynamics. We further show that single-channel models are limited to intraspecific dynamics,while interspecific interactions require multi-channel extensions—explaining why Extended Lenia’s multiplechannels produce richer phenomena. Certain discrete rules (e.g., HighLife B36/S23) have disconnected survivalwindows no smooth function can produce, marking a boundary between continuous and discrete CA.
Franny Philos Sophia (Sun,) studied this question.